The extension of aggregated finite elements to indefinite systems has been analyzed in [1]}. We note that its application to stabilized finite element formulations that transform the indefinite system into a definite one is straightforward [2]}. The stability of indefinite problems relies on so-called inf-sup conditions. These conditions are stringent and are only valid for very specific i.e., finite element spaces. For instance, the velocity and pressure finite element spaces in the (Navier–)Stokes problem have to match to satisfy this condition. Aggregation on these spaces does not generally preserve inf-sup stability. The aggregated mixed finite element method for the Stokes problem in [1]} starts with the inf-sup stable pair \(Q_p(\mathcal {T}_h^{\textrm {in}}) \times \mathcal {P}_{p-1}^{-}(\mathcal {T}_h^{\textrm {in}})\) , i.e., the velocity resides component-wise in \(\mathcal {Q}_p(\mathcal {T}_h^{\textrm {in}})\cap {C}^0(\mathcal {T}_h^{\textrm {in}})\)  and the pressure is a discontinuous function
in \(\mathcal {P}_{p-1}(\mathcal {T}_h^{\textrm {in}})\) . The extension defined above for continuous and discontinuous spaces leads to a pair that does not satisfy the inf-sup condition. Instead, a modified extension of the velocity components is proposed in [1]}, which relies on serendipity finite elements. In some specific situations, additional pressure jump stabilization terms must also be added.
